Consider the family: $$ \begin{matrix} X & = & \textbf{Proj}\left(\frac{\mathbb{C}[s,t,u][x,y,z]}{s(x^4 -y^2z^2) + t(x^2y^2 - z^4) - u(x^4 + y^4 + z^4)}\right) & \\ \downarrow & & \downarrow\\ B & = & \textbf{Proj}(\mathbb{C}[s,t,u]) \end{matrix} $$ How can I find the hyperplane $H_x \subset B$ of points whose fibers contain a point $x \in X$? For example, let $x = [0:1:1:1:(-1)^{(1/3)}:(-1)^{(1/3)}]$. For reference, I am looking at Nicolaescu's introduction to morse theory (page 252).

For reference, here is what I have tried: if I have the point $[s:t:u:a:b:c] \in X$, then I think the hyperplane will be spanned by the set of points $$ \{s,t,u \in \mathbb{C}: s(a^4 - b^2c^2) + t(a^2b^2 - c^4) + u(a^4 + b^4 + c^4)\} $$ I am guessing this because of the statement afterwards $$ H_x = \{ P \in U : P(x) = 0 \} $$ where $U$ is the projective plane parametrizing this family, but I am not sure why this is true.

  • $\begingroup$ If someone has enough reputation, can they add a linear system tag? $\endgroup$ – 54321user Aug 4 '16 at 23:51
  • $\begingroup$ There is only a nonlinear system tag. $\endgroup$ – 关一骏 Aug 15 '16 at 0:47
  • $\begingroup$ If that is the right one, will do. Actually, I think it has already been done by someone else. $\endgroup$ – 关一骏 Aug 15 '16 at 0:47
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    $\begingroup$ Nonlinear systems are a far distance from algebraic geometry... $\endgroup$ – 54321user Aug 15 '16 at 5:21
  • $\begingroup$ In fact, the more appropriate tag is divisors. I'll add it now. $\endgroup$ – paf Aug 15 '16 at 17:46

If the fiber over $[s:t:u]\in B$ contains $x=[s':t':u':a:b:c]\in X$ then this means that, denoting by $P$ the equation of $X$ you wrote :

$$[s:t:u]=[s':t':u']\text{ and }P(s',t',u',a,b,c)=0.$$

In fact, by definition of $X\to B$, the fiber over $[s_0:t_0:u_0]\in B$ is $$\{[x:y:z]\in \Bbb P^2(\Bbb C) : P(s_0,t_0,u_0,x,y,z)=0\}\subseteq \mathrm{Proj}(\Bbb C[s_0,t_0,u_0][x,y,z])\subseteq \mathrm{Proj}(\Bbb C[s,t,u][x,y,z]).$$

So, what you wrote is almost correct : I would write $H_x=\{ [s:t:u]\in \Bbb P^2(\Bbb C) : s(a^4 - b^2c^2) + t(a^2b^2 - c^4) + u(a^4 + b^4 + c^4)\}$ instead of $\{ s,t,u\in\Bbb C : \dots\}$.

Edit: Now, denoting by $\hat{U}$ the dual projective space of $U$, if we want to describe explicitly the map $X\to \hat{U}$ defined by $x\mapsto H_x$, we can write it in coordinates. For this purpose, note that if $(As+Bt+Cu=0)$ is the equation of a hyperplane $H$ of $\text{Proj}(\Bbb C[s,t,u])$, then the coordinates of $H$ in $\hat{U}$ are $[A:B:C]$.

Hence, in your example, the map $x\mapsto H_x$ is defined by $$[s:t:u:a:b:c]\mapsto [a^4 - b^2c^2:a^2b^2 - c^4:a^4 + b^4 + c^4].$$

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  • $\begingroup$ This doesn't entirely answer the part of the question I raised in the bounty message. $\endgroup$ – 54321user Aug 13 '16 at 19:15
  • $\begingroup$ In fact, I don't understand very well what you wrote here. What is "the vanishing locus of the hyperplanes"? (i.e. what does mean "vanishing" for a hyperplane?) Moreover, I don't know the definition of the "modification" of a linear system (and I don't have Nicolaescu's notes). $\endgroup$ – paf Aug 13 '16 at 19:32
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    $\begingroup$ This is on page 212 of www3.nd.edu/~lnicolae/Morse2nd.pdf $\endgroup$ – 54321user Aug 13 '16 at 19:44
  • $\begingroup$ Ok, thanks, and what's your problem with the definition of "modification"? $\endgroup$ – paf Aug 13 '16 at 20:06
  • $\begingroup$ There is a lot of jumping around with information. I think the computation I was doing above was computing a correspondence between $X$ and $U$ not $\hat{U}$. I'm not sure how to figure out this morphism to the dual variety. $\endgroup$ – 54321user Aug 13 '16 at 20:28

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